### Menger Algebras and Clones of Terms Klaus Denecke

#### Abstract

Clones are sets of operations which are closed under superposition

and contain all projections. The superposition operation maps to each

(n+1)?tuple of n-ary operations a new n-ary operation and satisfies

the so-called superassociative law. The corresponding algebraic structures are Menger algebras of rank n, unitary Menger algebras of rank

n and Menger algebras with infinitely many nullary operations. Identities

of clones of term operations of a given algebra correspond to hyperidentities of this algebra, i.e. to identities which are satisfied after any

replacements of fundamental operations by derived operations ([10]). If

any identity of an algebra is satisfied as a hyperidentity, the algebra is

called solid ([3]). Solid algebras correspond to free clones. These connections will be extended tostrongly full clones, togeneralized clones,

to strong hyperidentities and to strongly solidvarieties. We prove that n hyperidentities, SF-hyperidentities and strong hyperidentities correspond

to identities in free unitary Menger algebras of finite rank, in Menger algebras of finite rank or to free unitary Menger algebras with infinitely

many nullary operations, respectively.

and contain all projections. The superposition operation maps to each

(n+1)?tuple of n-ary operations a new n-ary operation and satisfies

the so-called superassociative law. The corresponding algebraic structures are Menger algebras of rank n, unitary Menger algebras of rank

n and Menger algebras with infinitely many nullary operations. Identities

of clones of term operations of a given algebra correspond to hyperidentities of this algebra, i.e. to identities which are satisfied after any

replacements of fundamental operations by derived operations ([10]). If

any identity of an algebra is satisfied as a hyperidentity, the algebra is

called solid ([3]). Solid algebras correspond to free clones. These connections will be extended tostrongly full clones, togeneralized clones,

to strong hyperidentities and to strongly solidvarieties. We prove that n hyperidentities, SF-hyperidentities and strong hyperidentities correspond

to identities in free unitary Menger algebras of finite rank, in Menger algebras of finite rank or to free unitary Menger algebras with infinitely

many nullary operations, respectively.

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